# Fresnel coefficients and Fabry-Perot formula for spatially dispersive metallic layers

## TL;DR

This paper derives analytic formulas for the optical response of spatially dispersive metallic layers using a hydrodynamic model, simplifying predictions for multilayer structures and explaining the role of dielectric layers in reproducing nonlocal effects.

## Contribution

It introduces a simplified analytic approach for modeling spatially dispersive metals using a Fabry-Perot formalism, enabling accurate predictions with modified Fresnel coefficients.

## Key findings

- Modified Fresnel coefficients accurately predict multilayer responses
- Simplified formulas preserve accuracy while reducing complexity
- Adding dielectric layers reproduces nonlocal effects in multilayers

## Abstract

The repulsion between free electrons inside a metal makes its optical response spatially dispersive, so that it is not described by Drude's model but by a hydrodynamic model. We give here fully analytic results for a metallic slab in this framework, thanks to a two-modes cavity formalism leading to a Fabry-Perot formula, and show that a simplification can be made that preserves the accuracy of the results while allowing much simpler analytic expressions. For metallic layers thicker than 2.7 nm modified Fresnel coefficients can actually be used to accurately predict the response of any multilayer with spatially dispersive metals (for reflection, transmission or the guided modes). Finally, this explains why adding a small dielectric layer[Y. Luo et al., Phys. Rev. Lett. 111, 093901 (2013)] allows to reproduce the effects of nonlocality in many cases, and especially for multilayers.